Using the fact that Let p be a prime number and let a1 a2

Using the fact that:

Let p be a prime number and let a1, a2, . . . , an, n 2, be integers such that p | a1a2 · · · an. Then p | ai for some index i, 1 i n

Prove that if a^k = p(b^k) then p | gcd(a,b)

Solution

a^k=pb^k

Hence, p|a using the fact given in the problem. So , a=mp for some integer m

So,

(mp)^k=pb^k

m^kp^{k-1}=b^k

Hence, p|b^k and again using the given fact

p|b

So, p|a and p|b

HEnce, p| gcd(a,b)